# division

Division is the inverse operation of multiplication defined so that if

*a* × *b* = *c*,

where *b* is nonzero, then

*a* = *c* /*b*

In this equation, *a* is the quotient, *b* is the **divisor**, and *c* is the **dividend**.
If dividend and divisor have like positive or negative sign, then the quotient
is positive; if their signs are different, the quotient is negative,

e.g., 6/3 = –6/–3 = 2, and –6/3 = 6/–3 = –2.

Division involving two powers of a number or variable is performed by subtracting the exponent of the divisor from that of the dividend:

*x*^{ 4} - *x *^{2} = *x *^{4-2} = *x *^{2};

the division of the coefficients being carried out in the same way:

*ax *^{7}/*bx *^{9} = (*a*/*b*)*x*^{-2}.

In the division of integers, where an integral quotient is required, it may well be that the division does not divide exactly into the dividend: thus 7/3 = 2 remainder 1 (or, in real numbers, 2 1/3).

A **skeletal division** is a long
division in which most or all of the digits are replaced by a symbol (usually
asterisks) to form a cryptarithm.

## Division of a polynomial by a binomial

Division of a polynomial by a binomial can be achieved by factoring if the binomial is a factor of the polynomial. If not, a more complex procedure is used.

To divide a polynomial by a monomial, divide each term separately by the monomial. Thus:

*ax*

^{ 4}

*y*

^{ 2}+

*bxy*

^{3}–

*cx*divided by –

*dxy*is

- (*a*/*d*)*x*^{ 3}*y* -
(*b*/*d*)*y *^{2} + *c*/*d*(*y*^{–1})

## Glossary of division terms

### Obelus

An obelus is the symbol "÷",
used as a sign for division. The term comes from
the Greek *obelos* meaning a pointed stick – a spit –
used for cooking. This root word also gave rise to "obelisk" for a pointed
stone pillar of stone. The "÷"
symbol was originally used as an editing mark in early manuscripts, sometimes
only as a line without the two dots, to point out material that the editor
thought needed cutting. It was also used occasionally as a symbol for subtraction.
As a division symbol it was first employed by the Swiss mathematician Johann
Rahn (1622–1676) in his *Teutsche Algebra* in 1659. By a misunderstanding
of a credit to John Pell about other material in the book, many English
writers started using the symbol and calling it **Pell's notation**.
Although it appears regularly in literature produced in Britain and the
United States, it is virtually unknown in the rest of the world.

### Quotient

A quotient is the number of times that one number can be divided exactly into another.
In the division of one positive integer by another, the quotient is the largest number of times (*k*) the
divisor (*a*) must be multiplied so that 0 < or = (*b* – *ka*)
< *a*, where *b* is the dividend and (*b* – *ka*)
the remainder.